Difference between revisions of "2005 AMC 12A Problems/Problem 4"
(→Solution) |
Sevenoptimus (talk | contribs) m (Fixed typo in problem statement) |
||
| (2 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
== Problem == | == Problem == | ||
| − | A store normally sells windows at | + | A store normally sells windows at <math>\$100</math> each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately? |
<math> | <math> | ||
(\mathrm {A}) \ 100 \qquad (\mathrm {B}) \ 200 \qquad (\mathrm {C})\ 300 \qquad (\mathrm {D}) \ 400 \qquad (\mathrm {E})\ 500 | (\mathrm {A}) \ 100 \qquad (\mathrm {B}) \ 200 \qquad (\mathrm {C})\ 300 \qquad (\mathrm {D}) \ 400 \qquad (\mathrm {E})\ 500 | ||
</math> | </math> | ||
| + | |||
== Solution == | == Solution == | ||
For <math>n</math> windows, the store offers a discount of <math>100 \cdot \left\lfloor\frac{n}{5}\right\rfloor</math> ([[floor function]]). Dave receives a discount of <math>100 \cdot \left\lfloor \frac{7}{5}\right \rfloor = 100</math> and Doug receives a discount of <math>100 \cdot \left\lfloor \frac{8}{5}\right\rfloor = 100</math>. These amount to <math>200</math> dollars in discounts. Together, they receive a discount of <math>100 \cdot \left\lfloor \frac{15}{5} \right\rfloor = 300</math>, so they save <math>300-200=100\ \mathrm{(A)}</math>. | For <math>n</math> windows, the store offers a discount of <math>100 \cdot \left\lfloor\frac{n}{5}\right\rfloor</math> ([[floor function]]). Dave receives a discount of <math>100 \cdot \left\lfloor \frac{7}{5}\right \rfloor = 100</math> and Doug receives a discount of <math>100 \cdot \left\lfloor \frac{8}{5}\right\rfloor = 100</math>. These amount to <math>200</math> dollars in discounts. Together, they receive a discount of <math>100 \cdot \left\lfloor \frac{15}{5} \right\rfloor = 300</math>, so they save <math>300-200=100\ \mathrm{(A)}</math>. | ||
| Line 12: | Line 13: | ||
[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 14:19, 1 July 2025
Problem
A store normally sells windows at 
each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately?
Solution
For 
windows, the store offers a discount of 
(floor function). Dave receives a discount of 
and Doug receives a discount of 
. These amount to 
dollars in discounts. Together, they receive a discount of 
, so they save 
.
See also
| 2005 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 3 |
Followed by Problem 5 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
